You on the right track.
Swapping pin 5 to pin 6 is the first thing you needed to do. Basically this will provide power to your shutter when the high beam is triggered. Coding will ensure the power is going to pin 6 when you flash and when you want to trigger the high beam.
The last mod you need to do is getting the ballast grounded. So it looks like your pin 2 and 4 are used. This means that pin 2 is most likely the ground for your ballast and pin 4 is the ground for some other lights in the headlight. your cars harness does not use pin 2 so you will need to tap the wires from pin 2 to pin 4. If it is tight, you can do it externally on the car wiring harness by making a pin and connecting it to the empty pin 2 slot and then tapping it to pin 4 wire externally.
I don't think swapping pin 2 and 4 is going to work as it may be providing the ground (most likely) for your angel eyes.
Another thing you may want to do is providing power to those 3 led lights on the side of the headlight near the top. Those probably go to a pin that isn't set up. You may want to the positive to the positive wire for your angel eyes.
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